2014/02/28 by Robert I. McLachlan, Klas Modin, Olivier Verdier
Engineering · Mathematics · Physics and Astronomy · #Advanced Numerical Methods in Computational Mathematics #Cotangent bundle #Equivariant map #Geodesic #Geometry #Hamiltonian (control theory) #Integrator #Mathematical analysis #Mathematics #Midpoint #Model Reduction and Neural Networks #Numerical methods for differential equations #Physics #Pure mathematics #Symplectic geometry #Symplectic integrator #Symplectic manifold #Symplectic representation #Symplectic vector space #Trigonometric functions #math-ph #math.MP #msc:37M15 #msc:53Z05 #msc:65L06 #msc:70H06 #msc:70H08
paper · pdf · doi:10.1090/mcom/3153
published as Mathematics of Computation, Vol. 86, No. 307, pp. 2325-2344 (2017)
openalex publication_date 2016/04/14 · arxiv created 2017/05/18 · arxiv updated 2017/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We construct a symplectic, globally defined, minimal-variable, equivariant integrator on products of 2-spheres. Examples of corresponding Hamiltonian systems, called spin systems, include the reduced free rigid body, the motion of point vortices on a sphere, and the classical Heisenberg spin chain, a spatial discretisation of the Landau–Lifshitz equation. The existence of such an integrator is remarkable, as the sphere is neither a vector space, nor a cotangent bundle, has no global coordinate chart, and its symplectic form is not even exact. Moreover, the formulation of the integrator is very simple, and resembles the geodesic midpoint method, although the latter is <italic>not</italic> symplectic.